Greatest Common Divisor (GCD) of 119 and 188
The greatest common divisor (GCD) of 119 and 188 is 1.
What is the Greatest Common Divisor (GCD)?
The GCD of two integers is the largest positive integer that divides both numbers without leaving a remainder. It is useful in simplifying fractions, finding common factors, and in number theory.
How to Calculate the GCD of 119 and 188?
We use the Euclidean algorithm, which involves the following steps:
- Divide the larger number by the smaller number.
- Replace the larger number with the smaller number and the smaller number with the remainder from the division.
- Repeat this process until the remainder is zero.
- The non-zero remainder just before zero is the GCD.
Step-by-Step Euclidean Algorithm
| Step | Calculation |
|---|---|
| 1 | 119 ÷ 188 = 0 remainder 119 |
| 2 | 188 ÷ 119 = 1 remainder 69 |
| 3 | 119 ÷ 69 = 1 remainder 50 |
| 4 | 69 ÷ 50 = 1 remainder 19 |
| 5 | 50 ÷ 19 = 2 remainder 12 |
| 6 | 19 ÷ 12 = 1 remainder 7 |
| 7 | 12 ÷ 7 = 1 remainder 5 |
| 8 | 7 ÷ 5 = 1 remainder 2 |
| 9 | 5 ÷ 2 = 2 remainder 1 |
| 10 | 2 ÷ 1 = 2 remainder 0 |
Examples of GCD Calculations
| Numbers | GCD |
|---|---|
| 89 and 140 | 1 |
| 35 and 46 | 1 |
| 72 and 57 | 3 |
| 38 and 173 | 1 |
| 62 and 153 | 1 |